\(\int \frac {\log (1-\frac {g (c+d x)}{a+b x})}{(a+b x) (c+d x)} \, dx\) [257]

Optimal result

Integrand size = 33, antiderivative size = 27 \[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {g (c+d x)}{a+b x}\right )}{b c-a d} \] Output:

polylog(2,g*(d*x+c)/(b*x+a))/(-a*d+b*c)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(375\) vs. \(2(27)=54\).

Time = 0.35 (sec) , antiderivative size = 375, normalized size of antiderivative = 13.89 \[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {-\log ^2\left (\frac {a}{b}+x\right )+2 \log \left (\frac {a}{b}+x\right ) \log (a+b x)-2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log (a+b x)+2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log \left (\frac {(b-d g) (a+b x)}{(b c-a d) g}\right )-2 \log \left (\frac {a}{b}+x\right ) \log (c+d x)+2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log (c+d x)+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log \left (\frac {(b-d g) (c+d x)}{b c-a d}\right )+2 \log (a+b x) \log \left (\frac {a-c g+b x-d g x}{a+b x}\right )-2 \log (c+d x) \log \left (\frac {a-c g+b x-d g x}{a+b x}\right )+2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b (a-c g+b x-d g x)}{(b c-a d) g}\right )-2 \operatorname {PolyLog}\left (2,-\frac {d (-a+c g-b x+d g x)}{-b c+a d}\right )}{2 b c-2 a d} \] Input:

Integrate[Log[1 - (g*(c + d*x))/(a + b*x)]/((a + b*x)*(c + d*x)),x]
 

Output:

(-Log[a/b + x]^2 + 2*Log[a/b + x]*Log[a + b*x] - 2*Log[(a - c*g)/(b - d*g) 
 + x]*Log[a + b*x] + 2*Log[(a - c*g)/(b - d*g) + x]*Log[((b - d*g)*(a + b* 
x))/((b*c - a*d)*g)] - 2*Log[a/b + x]*Log[c + d*x] + 2*Log[(a - c*g)/(b - 
d*g) + x]*Log[c + d*x] + 2*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] - 2 
*Log[(a - c*g)/(b - d*g) + x]*Log[((b - d*g)*(c + d*x))/(b*c - a*d)] + 2*L 
og[a + b*x]*Log[(a - c*g + b*x - d*g*x)/(a + b*x)] - 2*Log[c + d*x]*Log[(a 
 - c*g + b*x - d*g*x)/(a + b*x)] + 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a* 
d)] + 2*PolyLog[2, -((b*(a - c*g + b*x - d*g*x))/((b*c - a*d)*g))] - 2*Pol 
yLog[2, -((d*(-a + c*g - b*x + d*g*x))/(-(b*c) + a*d))])/(2*b*c - 2*a*d)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2997, 2965, 2752}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx\)

\(\Big \downarrow \) 2997

\(\displaystyle \int \frac {\log \left (\frac {a+x (b-d g)-c g}{a+b x}\right )}{(a+b x) (c+d x)}dx\)

\(\Big \downarrow \) 2965

\(\displaystyle \int \frac {\log \left (\frac {a+x (b-d g)-c g}{a+b x}\right )}{-\frac {(b c-a d) (a+x (b-d g)-c g)}{a+b x}-a d+b c}d\frac {a+x (b-d g)-c g}{a+b x}\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {\operatorname {PolyLog}\left (2,1-\frac {a-c g+(b-d g) x}{a+b x}\right )}{b c-a d}\)

Input:

Int[Log[1 - (g*(c + d*x))/(a + b*x)]/((a + b*x)*(c + d*x)),x]
 

Output:

PolyLog[2, 1 - (a - c*g + (b - d*g)*x)/(a + b*x)]/(b*c - a*d)
 

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2965

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(q + 1)*(i/d)^q   Subst[Int[(b*f - a*g - (d*f - c*g)* 
x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d* 
x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && IntegersQ[m, q] && IGtQ[p, 0] && EqQ[d*h - c*i, 0]
 

rule 2997

Int[Log[(e_.)*((f_.)*((g_) + (v_.)/(w_)))^(r_.)]^(s_.)*(u_.), x_Symbol] :> 
Int[u*Log[e*(f*(ExpandToSum[v + g*w, x]/ExpandToSum[w, x]))^r]^s, x] /; Fre 
eQ[{e, f, g, r, s}, x] && LinearQ[w, x] && (FreeQ[v, x] || LinearQ[v, x]) & 
& AlgebraicFunctionQ[u, x]
 

Maple [A] (verified)

Time = 2.79 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(-\frac {\operatorname {dilog}\left (\frac {a d g -b c g}{b \left (b x +a \right )}+\frac {-d g +b}{b}\right )}{d a -b c}\)	\(46\)
default	\(-\frac {\operatorname {dilog}\left (\frac {a d g -b c g}{b \left (b x +a \right )}+\frac {-d g +b}{b}\right )}{d a -b c}\)	\(46\)
risch	\(-\frac {\operatorname {dilog}\left (\frac {a d g -b c g}{b \left (b x +a \right )}+\frac {-d g +b}{b}\right )}{d a -b c}\)	\(46\)
parts	\(\frac {\ln \left (1-\frac {g \left (d x +c \right )}{b x +a}\right ) \ln \left (d x +c \right )}{d a -b c}-\frac {\ln \left (1-\frac {g \left (d x +c \right )}{b x +a}\right ) \ln \left (b x +a \right )}{d a -b c}-\frac {g \left (-\frac {\left (\frac {\operatorname {dilog}\left (\frac {\left (-d g +b \right ) \left (b x +a \right )+a d g -b c g}{a d g -b c g}\right )}{-d g +b}+\frac {\ln \left (b x +a \right ) \ln \left (\frac {\left (-d g +b \right ) \left (b x +a \right )+a d g -b c g}{a d g -b c g}\right )}{-d g +b}\right ) b \left (-d g +b \right )}{g \left (d a -b c \right )}+\frac {\ln \left (b x +a \right )^{2} b}{2 g \left (d a -b c \right )}\right )}{b}+\frac {g \left (-\frac {\left (\frac {\operatorname {dilog}\left (\frac {\left (-d g +b \right ) \left (d x +c \right )+d a -b c}{d a -b c}\right )}{-d g +b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {\left (-d g +b \right ) \left (d x +c \right )+d a -b c}{d a -b c}\right )}{-d g +b}\right ) d \left (-d g +b \right )}{\left (d a -b c \right ) g}+\frac {\left (\frac {\operatorname {dilog}\left (\frac {d a -b c +b \left (d x +c \right )}{d a -b c}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {d a -b c +b \left (d x +c \right )}{d a -b c}\right )}{b}\right ) d b}{\left (d a -b c \right ) g}\right )}{d}\)	\(425\)

Input:

int(ln(1-g*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-dilog((a*d*g-b*c*g)/b/(b*x+a)+(-d*g+b)/b)/(a*d-b*c)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {{\rm Li}_2\left (\frac {c g + {\left (d g - b\right )} x - a}{b x + a} + 1\right )}{b c - a d} \] Input:

integrate(log(1-g*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="fricas")
 

Output:

dilog((c*g + (d*g - b)*x - a)/(b*x + a) + 1)/(b*c - a*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(ln(1-g*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (26) = 52\).

Time = 0.05 (sec) , antiderivative size = 336, normalized size of antiderivative = 12.44 \[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx={\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (-\frac {{\left (d x + c\right )} g}{b x + a} + 1\right ) + \frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right )}{2 \, {\left (b c - a d\right )}} - \frac {\log \left (b x + a\right ) \log \left (\frac {{\left (d g - b\right )} a + {\left (b d g - b^{2}\right )} x}{b c g - a d g} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (d g - b\right )} a + {\left (b d g - b^{2}\right )} x}{b c g - a d g}\right )}{b c - a d} + \frac {\log \left (d x + c\right ) \log \left (\frac {c d g - b c + {\left (d^{2} g - b d\right )} x}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {c d g - b c + {\left (d^{2} g - b d\right )} x}{b c - a d}\right )}{b c - a d} + \frac {\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )}{b c - a d} \] Input:

integrate(log(1-g*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="maxima")
 

Output:

(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log(-(d*x + c)*g/(b* 
x + a) + 1) + 1/2*(log(b*x + a)^2 - 2*log(b*x + a)*log(d*x + c))/(b*c - a* 
d) - (log(b*x + a)*log(((d*g - b)*a + (b*d*g - b^2)*x)/(b*c*g - a*d*g) + 1 
) + dilog(-((d*g - b)*a + (b*d*g - b^2)*x)/(b*c*g - a*d*g)))/(b*c - a*d) + 
 (log(d*x + c)*log((c*d*g - b*c + (d^2*g - b*d)*x)/(b*c - a*d) + 1) + dilo 
g(-(c*d*g - b*c + (d^2*g - b*d)*x)/(b*c - a*d)))/(b*c - a*d) + (log(b*x + 
a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d))) 
/(b*c - a*d)
 

Giac [F]

\[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left (-\frac {{\left (d x + c\right )} g}{b x + a} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \] Input:

integrate(log(1-g*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="giac")
 

Output:

integrate(log(-(d*x + c)*g/(b*x + a) + 1)/((b*x + a)*(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\ln \left (1-\frac {g\,\left (c+d\,x\right )}{a+b\,x}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \] Input:

int(log(1 - (g*(c + d*x))/(a + b*x))/((a + b*x)*(c + d*x)),x)
 

Output:

int(log(1 - (g*(c + d*x))/(a + b*x))/((a + b*x)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\log \left (1-\frac {g (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\frac {-d g x +b x -c g +a}{b x +a}\right )}{-b \,d^{2} g \,x^{3}-a \,d^{2} g \,x^{2}+b^{2} d \,x^{3}-2 b c d g \,x^{2}+2 a b d \,x^{2}-2 a c d g x +b^{2} c \,x^{2}-b \,c^{2} g x +a^{2} d x +2 a b c x -a \,c^{2} g +a^{2} c}d x \right ) a^{2} d^{2} g -4 \left (\int \frac {\mathrm {log}\left (\frac {-d g x +b x -c g +a}{b x +a}\right )}{-b \,d^{2} g \,x^{3}-a \,d^{2} g \,x^{2}+b^{2} d \,x^{3}-2 b c d g \,x^{2}+2 a b d \,x^{2}-2 a c d g x +b^{2} c \,x^{2}-b \,c^{2} g x +a^{2} d x +2 a b c x -a \,c^{2} g +a^{2} c}d x \right ) a b c d g +2 \left (\int \frac {\mathrm {log}\left (\frac {-d g x +b x -c g +a}{b x +a}\right )}{-b \,d^{2} g \,x^{3}-a \,d^{2} g \,x^{2}+b^{2} d \,x^{3}-2 b c d g \,x^{2}+2 a b d \,x^{2}-2 a c d g x +b^{2} c \,x^{2}-b \,c^{2} g x +a^{2} d x +2 a b c x -a \,c^{2} g +a^{2} c}d x \right ) b^{2} c^{2} g -\mathrm {log}\left (\frac {-d g x +b x -c g +a}{b x +a}\right )^{2} b +\mathrm {log}\left (\frac {-d g x +b x -c g +a}{b x +a}\right )^{2} d g}{2 d g \left (a d -b c \right )} \] Input:

int(log(1-g*(d*x+c)/(b*x+a))/(b*x+a)/(d*x+c),x)
 

Output:

(2*int(log((a + b*x - c*g - d*g*x)/(a + b*x))/(a**2*c + a**2*d*x + 2*a*b*c 
*x + 2*a*b*d*x**2 - a*c**2*g - 2*a*c*d*g*x - a*d**2*g*x**2 + b**2*c*x**2 + 
 b**2*d*x**3 - b*c**2*g*x - 2*b*c*d*g*x**2 - b*d**2*g*x**3),x)*a**2*d**2*g 
 - 4*int(log((a + b*x - c*g - d*g*x)/(a + b*x))/(a**2*c + a**2*d*x + 2*a*b 
*c*x + 2*a*b*d*x**2 - a*c**2*g - 2*a*c*d*g*x - a*d**2*g*x**2 + b**2*c*x**2 
 + b**2*d*x**3 - b*c**2*g*x - 2*b*c*d*g*x**2 - b*d**2*g*x**3),x)*a*b*c*d*g 
 + 2*int(log((a + b*x - c*g - d*g*x)/(a + b*x))/(a**2*c + a**2*d*x + 2*a*b 
*c*x + 2*a*b*d*x**2 - a*c**2*g - 2*a*c*d*g*x - a*d**2*g*x**2 + b**2*c*x**2 
 + b**2*d*x**3 - b*c**2*g*x - 2*b*c*d*g*x**2 - b*d**2*g*x**3),x)*b**2*c**2 
*g - log((a + b*x - c*g - d*g*x)/(a + b*x))**2*b + log((a + b*x - c*g - d* 
g*x)/(a + b*x))**2*d*g)/(2*d*g*(a*d - b*c))